[seqfan] Re: Primes that are the sum of 4 distinct primes; and Primes that are not the sum of 4 distinct primes

[William Keith]

On May 16, 2010, at 7:22 PM, franktaw at netscape.net wrote:

> Do you have a reference for that? Naively, it appears to require 
> something a bit stronger than the Goldbach conjecture.
> 
> It also doesn't seem to quite imply the Goldbach conjecture: write 2n-2 
> as the sum of two primes p+q; maybe 2n is p+q+2 but not the sum of 2 
> distinct primes, and 2n+2 is the sum of two distinct primes but not of 
> three.
> 
> Franklin T. Adams-Watters

Same page, http://primes.utm.edu/notes/conjectures/ .  "Schnizel showed that Goldbach's conjecture is equivalent to every integer n > 17 is the sum of three distinct primes."  This should be Schinzel.

Another reference would be http://mathworld.wolfram.com/GoldbachConjecture.html : "Other variants of the Goldbach conjecture include the statements that every even number [at least] 6 is the sum of two odd primes, and every integer > 17 the sum of exactly three distinct primes."

www.emis.ams.org/journals/AMI/2004/bui.pdf includes the claim but not a ref.  A book by Schinzel himself ( http://books.google.com/books?id=ktCZ2MvgN3MC&lpg=PA123&ots=DrHBY7wnNT&dq=Schinzel%20Goldbach&pg=PA124#v=onepage&q=Schinzel%20Goldbach&f=false ) from 2005 says Goldbach implies this statement but does not say the reverse.  I also see a list of Schinzel's papers but his two on Goldbach are in French, which I can't read.

So I suppose your skepticism is warranted, and unless someone can verify the claim with an original paper somewhere it shouldn't be cited.

William Keith